With three fair dice abc every number is one in 6.6.6 = 216 but there are six combinations with the same outcome,

Abc,acb,bac,bca,cab,cba, so any pattern is a permutation achieved six ways, assuming the dice are equally important, but the order does not matter.

To score 3 the same is unique, 1,1,1 all three have that 1/6 chance so the probability is 1/216

Same with 666 but the numbers in between have several solutions such as 9 which can be made:

126,135,144,225,234,333,

If we take 126, 135,144 the chance of the first 1 is 1/6

But after the one, the second die can be 2,3,4 which is 3/6. The third die is then fixed and the chance is 1/6.

That makes 1/6*3/6*1/6= 1/72

But the order does not matter so we have 6/72 or 1/12.

Then we have 225 and 234.

To get 2 is 1/6 but the second can be 2 or 3, 2/6.

But again the third has to be right 1/6,

We have 1/6*2/6*1/6 and again six combinations so 1/18.

Finally 333 is 1/216

So the chance of score 9 is 1/12+1/18+1/216

Every score can be worked out that way.

The point is a number like 333 can only happen one way but 123 can happen 6 ways. To score 4 requires 112, which can be done 3 ways, so the chance is 3/216, that applies to combinationa aab, (aab,aba,baa) I hope i got that right!

Last edited by profloater; 2021-Jan-06 at 05:30 AM.
Reason: Added the correct 6.6.6=216

sicut vis videre esto

When we realize that patterns don't exist in the universe, they are a template that we hold to the universe to make sense of it, it all makes a lot more sense.

*Originally Posted by Ken G*